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Hyperbolas are one of the four types of conic sections, which also include circles, ellipses, and parabolas. Conic sections are the curves obtained by intersecting a double cone with a plane. The shape and type of intersection depend on the angle at which the plane cuts through the cone. Hyperbolas emerge when the plane intersects both nappes of the cone, forming two separate curves. Each curve is called a branch. Understanding how hyperbolas fit within the family of conic sections helps in identifying their properties and applications. These curves have unique features like open branches, which differ from the closed loops of ellipses or the parabola's single curve.

To transform the equation of a hyperbola, one needs to understand the standard form:

• Horizontal hyperbola: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \)
• Vertical hyperbola: \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \)

The given equation, \( y^2 - x^2 = 1 \), indicates a vertical hyperbola centered at the origin \((0,0)\).
To transform it, we perform a shift on the coordinates. This involves substituting \( x \) with \( (x+1) \) and \( y \) with \( (y+1) \). After completing this substitution, the new equation becomes \((y+1)^2 - (x+1)^2 = 1\).
Successfully transforming a hyperbola's equation lays the groundwork for finding its other features post-shift.

Coordinate shifts in the hyperbola's equation involve moving its center from the origin to a new location. This is often achieved by adjusting the equation to replace \( x \) and \( y \) with new expressions reflecting a lateral and vertical shift. In the given exercise, the hyperbola was moved 1 unit left and 1 unit down.
Thus, the original center at \((0,0)\) becomes \((-1, -1)\). This change is executed by substituting \( (x, y) \) with \( (x+1, y+1) \) in the equation. When you change the equation's terms like this, you effectively shift the entire graph without altering the hyperbola's inherent shape.
Understanding this concept is key for redrawing or recalculating a hyperbola's other properties after a shift. Such transformations are crucial for positioning the hyperbola according to a problem's needs or data.

Vertices and foci are vital elements in appreciating a hyperbola’s structure. The vertices are the closest points to the center along the transverse axis, while the foci are located along the same axis but at a greater distance from the center. Calculating these points involves understanding the hyperbola equation's components.
In the original equation \( y^2 - x^2 = 1 \), the vertices were at \( (0, \pm1) \). After applying our coordinate shift to \((-1, -1)\), the vertices shift to \((-1, 0)\) and \((-1, -2)\). The foci need to be recalculated with the distance formula, using \( c = \sqrt{a^2 + b^2} \). Since \( a^2 = 1 \) and \( b^2 = 1 \), we compute \( c = \sqrt{2} \).
Thus, the foci of the shifted hyperbola move to \( (-1, -1 \pm \sqrt{2}) \). Calculating these correctly ensures one can fully draft the hyperbola's structure in its new position. Asymptotes such as \( y = \pm x - 1 \), while different from vertices and foci, also provide critical information about the hyperbola’s layout and direction.